Let f(x)= |1+ sin^2 x, cos^2 x, sin 2 x; sin^2 x, 1+ cos^2 x, sin 2 x; sin^2 x, cos^2 x, 1+ sin 2 x|, x ∈[(π)/6, (π)/3]. If α and β respectively are…

Let $\displaystyle f(x)=\left|\begin{array}{ccc}1+\sin ^2 x & \cos ^2 x & \sin 2 x \\ \sin ^2 x & 1+\cos ^2 x & \sin 2 x \\ \sin ^2 x & \cos ^2 x & 1+\sin 2 x\end{array}\right|, \quad x \in\left[\frac{\pi}{6}, \frac{\pi}{3}\right] . \quad$ If $\displaystyle \alpha$ and $\displaystyle \beta$ respectively are the maximum and the minimum values of $\displaystyle f$, then